Here's an easy proof of the equivalence of the statement to CH (I think). One direction is just what Emil said. For the other direction:

Suppose continuum is $\geq \aleph_2$. Restrict attention to an $\aleph_2$-sized subset of $\mathbb{R}$, and suppose we had a relation with the properties you want on that subset. (Note: if there is a relation with those properties on $\mathbb{R}^2$ then it will retain those properties when we restrict to a smaller set.)
Take $\aleph_1$ many $x$-coordinates from this set; each of them only has countably many $y$'s that it gets paired with in the relation, so in total there are at most $\aleph_1$ many $y$'s that get paired with any of these $x$'s. So, take some $y$ which doesn't get paired with any of these $x$'s. (There will be such a $y$ because we have $\aleph_2$ many $y$'s in total.) This $y$ has at least $\aleph_1$ many $x$'s that it does not get paired with; and this contradicts the properties of the relation.